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Like here's the video:

https://www.khanacademy.org/math/arithmetic-home/multiply-divide/division-intro/v/division-1?modal=1

Explanation 1 is at 4:08 Explanation 2 is at 5:31

How are the two related?

bumblehumble
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    What do you mean "the number of threes in a number related to the number divided into three equal parts"? The number of $3$s in $1353$ is $2$ and the number divided into $3$ equal parts is $451$. Are you saying $2$ related to $451$? How so? – fleablood Jul 09 '19 at 20:44
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    @fleablood I think the OP means that the number of threes in $n$ in ${n\over3}$ – saulspatz Jul 09 '19 at 20:46
  • Perhaps you are asking for the word quotient? – hardmath Jul 09 '19 at 20:52
  • So ... the number of "$3$s" in $51$ is $17$? Is it not clear that every third number is "a three"? ANd isn't $\frac {51}3$ means how many sets of three numbers "go into" $51$ not self-evident? – fleablood Jul 09 '19 at 20:53
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    This may help you: https://math.stackexchange.com/questions/1127483/how-to-make-sense-of-fractions/1127776#1127776 – Ethan Bolker Jul 09 '19 at 21:04

2 Answers2

1

Okay you are asking why splitting $51$ into $3$ groups will give you $17$ in each group and why splitting $51$ into groups of $3$ will give you $17$ groups of $3$. an both of them are the two different meanings of $51\div 3$?

Well that is because if you have $3$ groups of $17$ watermelons you will have $51$ watermelons and if you have $17$ groups of $3$ watermelons you will have $51$ watermelons.

So your question is why is $N$ groups of $k$ things the same thing as $k$ groups of $N$ things.

Well, the reason that is is:

Suppose you have $17$ groups of $3$ things. So you have $17$ bags of watermelons each with $3$ watermelons. If you take one watermelon from each bag and put them in a crate, you will have a crate of $17$ watermelons (one from each bag). Each bag will have $2$ watermelons left. You can do that again and have a second crate of $17$ watermelons. Now each bag has $1$ watermelon in it. Do it one last time and you get a third crate of $17$ watermelons and all the bags are empty.

So you have $3$ crates with each crate having $17$ watermelons.

So $17$ bags of $3$ watermelons each $=$ $3$ crates of $17$ watermelons each.

So $N$ groups of $k$ things = $k$ groups of $N$ things = $N\times k$ things.

And the value $51 \div 3$ can be thought of as either: If I split into $3$ groups how much is in each group. Or it can be thought of as: If I split this into groups of $3$ how many groups will I have.

These are the same because

$3$ groups with $?????$ each adding up to $51$

is the same thing as

$?????$ groups with $3$ each adding up to $51$.

jvdhooft
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fleablood
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The answer as a slogan is "Because multiplication is commutative." In other words, you can multiply numbers in any order.

For example, if you say that there are five threes in $15$, then we write that as the equation $$15=3+3+3+3+3=5\times3$$ But $5\times3=3\times5$, so we also know: $$15=3\times5=5+5+5$$ And so $15$ divided into three equal parts is $$15\div3=5$$

This also works with fractions. You can say that $1=\frac13\times3$, meaning that there is one-third of a $3$ in a $1$. This is equivalent to $1=3\times\frac13$, meaning that there are three one-thirds in a $1$.

Now, if you want to know why order doesn't matter in multiplication, that's another topic. But as a hint, it has to do with rotating rectangles.

Chris Culter
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  • By the way, I didn't watch the Khan academy video, so I don't know the degree to which my explanation overlaps with theirs. – Chris Culter Jul 09 '19 at 20:58